Covariant phase observables are obtained by defining simple conditions for mappings from the set of phase wave functions (unit vectors of the Hardy space) to the set of phase probability densities. The existence of phase probability density for any phase wave function, the existence of interference effects, and the natural phase shift covariance are those simple conditions. The nonlocalizability of covariant phase observables is proved.
Determining the measurement uncertainty region is a difficult problem for generic sets of observables. For this reason the literature on exact measurement uncertainty regions is focused on symmetric sets of observables, where the symmetries are used to simplify the calculation. We provide a framework to systematically exploit available symmetries, formulated in terms of covariance, to simplify problems of measurement uncertainty. Our key result is that for a wide range figures of merit the optimal compatible approximations of covariant target observables are themselves covariant. This substant…
We discuss symmetric quantum measurements and the associated covariant observables modelled, respectively, as instruments and positive-operator-valued measures. The emphasis of this work are the optimality properties of the measurements, namely, extremality, informational completeness, and the rank-1 property which contrast the complementary class of (rank-1) projection-valued measures. The first half of this work concentrates solely on finite-outcome measurements symmetric w.r.t. finite groups where we derive exhaustive characterizations for the pointwise Kraus-operators of covariant instrume…
We obtain an explicit characterization of linear maps, in particular, quantum channels, which are covariant with respect to an irreducible representation ($U$) of a finite group ($G$), whenever $U \otimes U^c$ is simply reducible (with $U^c$ being the contragradient representation). Using the theory of group representations, we obtain the spectral decomposition of any such linear map. The eigenvalues and orthogonal projections arising in this decomposition are expressed entirely in terms of representation characteristics of the group $G$. This in turn yields necessary and sufficient conditions…
The imposition of symmetry upon the nature and structure of quantum observables has recently been extensively studied, with quantum reference frames playing a crucial role. In this paper, we extend this work to quantum transformations, giving quantitative results showing, in direct analogy to the case of observables, that a "large" reference frame is required for non-covariant channels to be well approximated by covariant ones. We apply our findings to the concrete setting of SU(2) symmetry.
We study non-commutative operator graphs generated by resolutions of identity covariant with respect to unitary representations of a compact group. Our main goal is searching for orthogonal projections which are anticliques (error-correcting codes) for such graphs. A special attention is paid to the covariance with respect to unitary representations of the circle group. We determine a tensor product structure in the space of representation under which the obtained anticliques are generated by entangled vectors.
Time-independent gauge transformations are implemented in the canonical formalism by the Gauss law which is not covariant. The covariant form of Gauss law is conceptually important for studying asymptotic properties of the gauge fields. For QED in $3+1$ dimensions, we have developed a formalism for treating the equations of motion (EOM) themselves as constraints, that is, constraints on states using Peierls' quantization. They generate spacetime dependent gauge transformations. We extend these results to the Maxwell-Chern-Simons (MCS) Lagrangian. The surprising result is that the covariant Gau…
We study positive kernels on $X\times X$, where $X$ is a set equipped with an action of a group, and taking values in the set of $\mathcal A$-sesquilinear forms on a (not necessarily Hilbert) module over a $C^*$-algebra $\mathcal A$. These maps are assumed to be covariant with respect to the group action on $X$ and a representation of the group in the set of invertible ($\mathcal A$-linear) module maps. We find necessary and sufficient conditions for extremality of such kernels in certain convex subsets of positive covariant kernels. Our focus is mainly on a particular example of these kernels…
We present in the article the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg-Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors under the Lorentz symmetry. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group $C^*$-algebra giving the algebra of observables. The key feature of the formulation is that it is not unitary but pseudo-un…
